Group for Research in Decision Analysis

# Matrix decomposition with positivity constraint: Applications to clustering and dimensionality reduction

## Ekaterina Smirnova – University of Wyoming, United States

Non negative matrix factorization (NMF) is a dimension reduction technique that has applications in text mining, clustering, image analysis, signal processing, bioinformatics microarray data analysis and many more. This technique is based on factorization of a rectangular matrix $$X$$ into a product of two non-negative rank $$r$$ matrices $$W$$ and $$H$$. The idea of this method is closely related to singular value decomposition (SVD), however an additional non negativity constraint enhances interpretability of obtained basis vectors. As opposed to SVD, NMF is a non-convex problem, which cannot be solved analytically. This means that there might be multiple local solutions satisfying the problem, and no knowledge of their relation to a global solution. Since its introduction in the middle of 1990s, the development of NMF technique was mainly directed towards the improvement of algorithmic approaches to find non-negative factors of a matrix $$X$$. Discussion of conditions that guarantee this decomposition, and feasility of multiple solutions in interpreting final results has received far less attention. In this talk, we concentrate on open issues with NMF factorization, its connection to clustering problems, and possible extensions to problems of predicting non-negative response based on high dimensional data.

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